1. Estimating the firing rate
Tahereh Toosi
IPM

2. Brief Review of Spike Train Analysis
10:30-11:30
12-13
Thursday, 31 Jan
Estimating the Firing Rate of Spike Trains
Tahereh Toosi
Introduction to Parameter Estimation
HaDi MaBouDi
Thursday, 7 Feb
Spike-Train Statistics
Ehsan Sabri
Entropy and Information Theory
Thursday, 14 Feb
Spike-Train Encoding and Decoding
Safura Rashid Shomali
Statistical models of neural data
Thursday, 21 Feb
An Introductory to Information Geometry of Spike Trains
Population coding: Ising model and GLM
Thursday, 28 Feb,
Works on Real Data!!!

3. Outline Extracting Spikes Spike Sorting Neuronal coding types
Estimating the firing rate
Optimizing the rate estimation
Summary

4. Neural recordings

5. Spike Sorting

6. [NeuroQuest]

7. Neural Coding Three Coding schemes Rate coding Temporal coding
Spike-count rate
Time-dependent firing rate
Temporal coding
Phase-of-firing code
Spike Latency codes
Population coding
Position coding

8. Estimating the firing rate
Methods for estimating the firing rate
PSTH
The Kernel Density Estimation
Methods for optimizing the rate estimation
Minimizing Mean Squared error (MISE)
Maximizing likelihood
“Analysis of Parallel Spike Trains”, Chapter 2 Estimating the Firing Rate, S. Shinomoto

9. Methods for estimating instantaneous rate
Challenges to rate estimation

10. PSTH Method Peristimulus time histogram
Methods for estimating the firing rate
Peristimulus time histogram

11. PSTH Method
Methods for estimating the firing rate

12. Kernel Density Function Method
Methods for estimating the firing rate
Kernel Density Function Method
Kernel features:
the normalization to unit area, f (t)dt = 1.
nonnegative,f (t) ≥ 0,
have a finite bandwidth defined by the variance that is normally finite, 2 = t2f (t)dt <∞,
symmetric, f (t) = f (−t).

13. Kernel Density Function Method
Methods for estimating the firing rate
Kernel Density Function Method

14. Methods for Optimizing the Rate Estimation

15. MISE Minimizing Mean Integrated Squared Error Assumption on r(t) :
Methods for optimizing rate estimation
MISE
Minimizing Mean Integrated Squared Error
Assumption on r(t) :
spikes are drawn from nonhomogeneous Possion process
[Shimazaki and Shinomoto, 2007]

16. Methods for optimizing rate estimation
MISE for PSTH

17. Methods for optimizing rate estimation
MISE : results

18. MISE for Kernel Density Function
Methods for optimizing rate estimation
MISE for Kernel Density Function

19. Methods for optimizing rate estimation
MISE : Comparison of the optimized PSTH and optimized kernel density estimator

20. Maximum likelihood Time- dependent Poisson process
Methods for optimizing rate estimation
Maximum likelihood
Time- dependent Poisson process
The rate-modulated Poisson process.
The probability for a spike to occur in each short interval δt is r(t)δt<< 1, and the probability of having no spike is 1− r(t)δt ≈ exp(−r(t)δt)

21. Methods for optimizing rate estimation
probability of having no spike from the time t1 to t2 :
Bayes rule gives the “inverse probability”:
the estimated rate becomes
in/sensitive to individual spike occurrences as β is large/small
flatness
The probability of having spikes at {ti} ≡ {t1, t2, , tNs} is given by the “marginal
likelihood function”:
obtaining the maximum a posteriori (MAP) estimate of the rate ˆr(t), so that their posterior distribution maximized.

22. Comparison of the optimized KD and Empirical Bayes rate estimators
Methods for optimizing rate estimation
Comparison of the optimized KD and Empirical Bayes rate estimators

23. Empirical Bayes method
Methods for optimizing rate estimation
Empirical Bayes method

24. Summary Neuronal activity is measured by the number of spikes
Challenges to grasping the time-varying rate of spike firing
Binsize of the time histogram
Bandwidth of the kernel smoother
Standard rate estimation tools, such as
the peri-stimulus time histogram (PSTH)
kernel density estimation
Optimization of rate estimation
MISE
Maximum likelihood